# Questions tagged [sheaf-theory]

A Sheaf $\mathcal F$ on a topological space $X$ captures local data $\mathcal F(U)$ given on open sets $U\subseteq X$ and how such data can be restricted to smaller open sets or glued together. In typical cases, $\mathcal F(U)$ is a set of functions defined on $U$ and an element of $\mathcal F(V)$, $V\subseteq U$ is obtained by restricting the domain and not all elements of $\mathcal F(U)$ can be obtained by restricting a global section $\in\mathcal F(X)$.

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### Computing Exts and projective covers

Let $\mathcal{P}$ be the category of perverse sheaves on $\mathbb{P}^1$ over the field $\mathbb{C}$, where strata are the point $Z = {0}$, and its complement $U$. Let $i: Z \rightarrow \mathbb{P}^1$ ...
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### Is the presheaf of continuous functions on a topological space a “complete presheaf”?

Is the presheaf of continuous functions $f:A\rightarrow B$ from a topological space $A$ to another topological space $B$ a "complete presheaf"? Can't find this, anyone have a reference?
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### Reference for “It is enough to specify a sheaf on a basis”?

The wikipedia article on sheaves says: It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space. ...
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### Obtaining a morphism of sheaves from a morphism of presheaves

I think I need some help on a problem about sheaf theory. Suppose that $f: X \rightarrow Y$ is a continuous map of topological spaces, $\mathscr{F}$ is a sheaf on $X$. Prove that the morphism of ...
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### Is the sheaf of locally constant functions flasque?

Quick question, is the sheaf of locally constant functions flasque?
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### Locally free sheaves on locally ringed spaces

One can define the notion of a locally free sheaf (of finite rank) on any locally ringed space. If you restrict to the category of (noetherian?) schemes, this category is equivalent to the category ...
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### Examples of surjective sheaf morphisms which are not surjective on sections

Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of ...
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### Isomorphism in derived category of coherent sheaves

this question might seem a bit special, but it came up at a crucial point of a proof I read and so I would be very obliged if someone could explain this to me: given is a smooth projective variety $X$...
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### On the definition of morphisms of ringed spaces

Suppose that $(X,\mathcal{O}_X),(Y,\mathcal{O}_Y)$ are two ringed spaces and $(f,f^{\sharp}):(X,\mathcal{O}_X)\longrightarrow(Y,\mathcal{O}_Y)$ is a morphism of these two ringed spaces.I wonder why we ...
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### Sheaf cohomology: what is it and where can I learn it?

As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter. ...
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### Sheafification and the unique decomposition of morphisms

My question is an attempt to understand the proof of lemma 6.17.3 (page 164) in the Stacks Project: http://www.math.columbia.edu/algebraic_geometry/stacks-git/book.pdf This lemma states: Let $F$ ...
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### Lifting sheaves from the special fibre to the generic fibre

Let $R$ be a complete discrete valuation ring and let $S = \operatorname{Spec}(R)$. Let $\mathcal{X}\to S$ be a complete, regular, flat, connected $S$-scheme of finite type whose fibres are smooth, ...
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### (pre)sheaves on path connected neigh

Presheaves are defined by using neighborhoods of a point. Is there a way to restrict this construction to path connected neighborhoods of points? What is the name of the object which assigns other ...
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### Sheaf cohomology

Let $\mathcal{F}$ be a sheaf over, say, a paracompact differentiable manifold $M$. Then to compute the cohomology $H(M,\mathcal{F})$ of $\mathcal{F}$, we can use any acyclic resolution or use Cech ...
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### etale space v. covering space

On the Wikipedia page Sheaf, under the section "The etale space of a sheaf," the author claims that the etale space of the sheaf of (continuous) sections of a continuous map $Y \to X$ is (homeomorphic ...
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### Cohomological dimension of a non-compact interval

Iversen, in "Cohomology of Sheaves," proves a number of theorems about the cohomological dimension of a locally compact space. In particular, if $\mathcal{F}$ is any sheaf on $\mathbb{R}^1$, it is ...
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### Failure of isomorphisms on stalks to arise from an isomorphism of sheaves

It is well-known (Hartshorne 2.1.1) that if $F$ and $G$ are sheaves on a space $X$, then $\phi:F\rightarrow G$ is an isomorphism if and only if the induced stalk map $\phi_p:F_p\rightarrow G_p$ is an ...
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### functoriality of derivations

I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$. Now, fiberwise it's all good. But I do not understand how to define ...
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### Presheaf which is not a sheaf — holomorphic functions which admit a holomorphic square root

I'm thinking about a problem in Ravi Vakil's algebraic geometry notes http://math.stanford.edu/~vakil/216blog/ (Exercise 3.2B) and I'm having trouble with the second part of the exercise as my ...
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### Criterion for local systems

Given a sheaf on say a manifold, such that all its stalks are isomorphic to a fixed finite dimensional vector-space. Is it true that the sheaf is a local system?
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### Hartshorne exercise about sheaves on $\mathbb{P}^1$

I've been stuck on Exercise II.1.21(e) from Hartshorne's book for quite a while. It concerns the projective line $\mathbb{P}^1$ over an algebraically closed field $k$: write $\mathscr{H}$ for the ...
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### Stalks of the tensor product presheaf of two sheaves

Let $(X, \mathscr{O})$ be a ringed space and $\mathscr{F}, \mathscr{G}$ be sheaves of $\mathscr{O}$-modules on $X$. Define $\mathscr{H}(U) = \mathscr{F}(U) \otimes_{\mathscr{O}(U)} \mathscr{G}(U)$. ...
I have been learning about sheaves and am thinking about the following problem. Let $F$ and $G$ be sheaves, say of abelian groups, on a space $X$. The sheaf $Hom(F, G)$ is defined by $Hom(F, G)(U)=Mor(... 1answer 272 views ### Subbundles and subsheaves Let Let$E \rightarrow X$be a vector bundle on a manifold$X$. Let$\cal E$be the sheaf of sections of$E$. Let$\cal F$be a subsheaf of$\cal E$, and let$F$be the etale space of$\cal F$. What ... 3answers 297 views ### Can every element in the stalk be represented by a section in the top space? Let$S$be a sheaf over$X$and$r$an element in$S_x$for some$x$in$X$. Must there exist a section$s$in$S(X)$such that such that$s$equals$r$when mapped to$S_x$by the canonical map? 0answers 5k views ### Pullback and Pushforward Isomorphism of Sheaves Suppose we have two schemes$X, Y$and a map$f\colon X\to Y$. Then we know that$\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where$\...
Suppose that $X$ is a topological space with a sheaf of rings $\mathcal{O}_X$. In general, the stalk at a point $p \in X$ is the direct limit of the rings $\mathcal{O}_X(U)$ for all open sets $U$ ...